How to create parallel edge between vertex and edge?
the Toolpanel (t-key) have no Join tool maybe you lock for.
But operator exist in the edge or vertex menü to solv your problem.
Mouse rightclick brings  a compact menü with operators
but depense on the select modes you choose (vertex,edge,or face) .
and left click or rightclick 2.8 Mode you use and  choose .

It could be that joining from a vertex to edge needs extra steps or actions.

So.
Select the edge thats needs the conection  with
 activ 'edge select Mode '.
and then 
rightclick with the mouse and  look and  click on  the  edge Menü operator 

'subdivide edge'.

This subdivide the  edge  and creats one new vertex in the middle of your selected edge .

Now use the vertex select Mode.
Select the  new created vertex on your edge  hold shift key and select the second vertex .
rightclick with mouse and find i
n the vertex  menu operator    called 'Join " .
That it.
a new edge conection between your vertex and edge(new vertex) is created that flows like you want.
Wfg